Deriving Lehmer and Hölder means as maximum weighted likelihood estimates for the multivariate exponential family

The links between the mean families of Lehmer and H\"older and the weighted maximum likelihood estimator have recently been established in the case of a regular univariate exponential family. In this article, we will extend the outcomes obtained to the multivariate case. This extension provides a probabilistic interpretation of these families of means and could therefore broaden their uses in various applications.